We saw yesterday that a sequence of geometric transformations can illustrate the same “fixed point attractor” behavior that we’ve seen when iterating a mathematical expression. We explained this by illustrating that such geometric transformations are analogous to iterating a linear function in the Complex number system. Specifically, adding two complex numbers is analogous to translating, and multiplying two complex numbers is analogous to a combined dilation/rotation.

The conventional rectangular form of a complex number a + bi tells us the horizontal and vertical components of the translation achieved by adding the complex number, but this form is not useful for when the number is being used as a multiplication factor. For that, we need the number’s Polar Form. Once finding it, the value of r tells us the dilation factor and the value of θ tell us the rotation factor.

We’ll dig in to this some more tomorrow, but for now please watch this video recapping polar vs rectangular coordinates (which the video refers to as “Cartesian coordinates”) and how to convert between the two.

We didn’t actually finish our lesson yesterday, so today’s HW is the same as yesterday’s. We did some more work with the quadratic formula today, specifically with simplifying the radical expressions we sometimes get when we apply it.

We wrapped up our conversation about numbers of cycles findable in the Feigenbaum Plot, noting again that for every attracting cycle we find, there is an accompanying “evil twin” repelling cycle. These “evil twin” repelling cycles serve a purpose. As the members of the attracting cycle bifurcate, their patterns branch out further and further. When this pattern of branching reaches one of the values of the repelling cycle – indeed it will reach all members of the repelling cycle simultaneously – the branching stops and the system dissolves into chaos once more. This explains why these spontaneously generating cycles appear in “windows” in the Feigenbaum Plot, with an abrupt end to chaos on the left, and an abrupt resuming of chaos on the right.

We wrapped up today with a recap of the work we’ve done so far, occupying our time exclusively within the Real numbers. The last portion of the course will now branch into the Complex numbers, effectively repeating the analyses we’ve done with iterating functions into the complex plane. Our first stop will be to understand how to represent geometric transformation with operations on complex numbers. We modeled this in class by showing how the repetition of a series of geometric transformations can result in the same “fixed point attracting” behavior as we saw with the real numbers.

Each week, starting today, we will be participating in the New York Times’s What’s Going On in This Graph activity. Click the link to see today’s graph, and go to Desmos to enter some of your thoughts.

We did some more work with the quadratic formula today, specifically with simplifying the radical expressions we sometimes get when we apply it.

You will spend this week working on your projects. You should strive to be done with data collection by Thursday the 12th, and you’ll have up to Monday the 16th to work on your presentations and written report. Refer to the contents of the midterm project Drive folder for details

We’ll spend the 17th-20th presenting on your findings, and your reports (one per group!) will be due on the 20th.

We have observed that cycles are born in the Feigenbaum Plot in one of two ways: bifurcations of lower cycles and spontaneously out of chaos. Yesterday we understood how these cycles are spontaneously born and how this phenomenon coupled with the self-similarity of the Feigenbaum Plot suggests an order to cycles. What it also gives us is a way of counting how many cycles of each type there are.

We know there are two fixed points on the original f (x) = ax(1 – x). One is attracting for a < 3, the other (at zero) is always repelling.

For 2-cycles, we look at f (f (x)). We see as many as four fixed points on this graph. But two are already members of 1-cycles and must be eliminated. This leaves only two points eligible to be members of 2-cycles, meaning we only have one 2-cycle, which we see bifurcating at a = 3.

To count 3-cycles, we observe that f (f (f (x))) has as many as eight fixed points. A 3-cycle pattern n, ___, ____, n could be explained two ways: as a 1-cycle (n, n, n, n) or as a real 3-cycle (n, o, p, n). So again, we remove the two 1-cycle points and are left with six points eligible to be members of 3-cycles, suggesting two 3-cycles. We see one in the Feigenbaum plot at a ≈ 3.83, but that’s an attracting fixed point. Where’s the other one? It turns out that for every cycle born out of chaos, there is an “evil twin” repelling cycle born as well. As a result, there are actually two 3-cycles born at a ≈ 3.83: one attracting and one repelling.

To count 4-cycles, we observe that f(f(f(f(x)))) has as many as 16 fixed points. But if we see a pattern of n, ___, ____, ____, n, we could explain this by:

A 1-cycle: n, n, n, n, n

A 2-cycle: n, o, n, o, n

A real 4-cycle: n, o, p, q, n

So we eliminate the two 1-cycle points and the two 2-cycle points. This leaves 12 points eligible to be members of 4-cycles, suggesting three 4-cycles. One is the bifurcating cycle we clearly see in the Feigenbaum plot, the second and third are an attracting/repelling 4-cycle pair found at a ≈ 3.96.

To count 5-cycles, we start with the 32 fixed points, eliminate the two that are members of 1-cycles, and observe that the 30 remaining points must create six 5-cycles. One attracting/repelling pair we see at a ≈ 3.74, but the other two are harder to find (one is at a ≈ 3.906, the other pair at a ≈ 3.99028).

Your homework for this weekend: continue this train of thought and identify how many 6-, 7-, and 8-cycles there are. If you’re feeling ambitious, try to locate all of them!

We observed yesterday that cycles appear to be “born” in one of two ways: bifurcations of “lower” cycles, and spontaneously arising from chaos. We’ve already shown why cycles bifurcate, so we started today with an explanation about how cycles spontaneously emerge from chaos.

Referring to our previous proof, graphing the function y = f (f (x)) can be a way of finding new cycles. Fixed points on the graph of y = f (f (x)) that are not common with the graph of y = f (x) will be the parameters of our 2-cycle. By extension, graphing y = f (f (f (x))), or any number of nested iterations, will give us a tool of finding new cycles. More crucially, this also shows us why cycles spontaneously appear from chaos. Explore the graph here. For a = 3.84, the “wiggliness” of the graph of y = f (f (f (x))) is enough for the fingers of the graph to touch the line y = x. But for a < 3.84, it isn’t. The moment that a becomes large enough for those fingers to touch the line y = x is the moment that a 3-cycle is born.

The graph above shows that it is not possible to get a 3-cycle before a = 3.84, meaning the 3-cycle “window” we see in the Feigenbaum Plot is the first time we get a 3-cycle (addressing one of the other questions we asked yesterday). It also gives a clue to the order of cycles. We’ve already noticed that the Feigenbaum Plot exhibits fractal-like self-similar behavior, and the 6-cycle we observed at a = 3.63 could almost be viewed as two groups of three. If we consider that the 3-cycle at a = 3.84 is “born” from the original fixed point trend we observed for a < 3.0, then we could argue that the 6-cycle is actually two conjoined 3-cycles, each born from the first bifurcation at a = 3.0. This would suggest that there is a 12-cycle for an even lower value of a, born from the second bifurcations 4-cycle (and indeed there is, at a = 3.5821).

The 5-cycle we see at a = 3.74 then is mirrored with a 10-cycle at a = 3.6053, and a 20-cycle at a = 3.5775. This pattern could continue forever, to find any cycle, of any length.

This argument forms the basis for the Sharkovskii order we saw in yesterday’s article. The 3 cycle is the very last cycle to be born out of the chaos of this trend. The 5-cycle is the second-to-last, and the 7-cycle and every other odd-numbered cycle comes before those. But before we get to any odd-numbered cycle, we first would find the 6-cycle (2 x 3). Before that, the 10 cycle (2 x 5); before that, the 14 -cycle (2 x 5), and so on. But before any of those, we find the 12-cycle (4 x 3); before that the 20-cycle (4 x 5); before that the 28-cycle (4 x 7). And so on, reading the Feigenbaum plot right-to-left, until we find our “un-bifurcating” powers of two cycles, stitching back together to 16, to 8, to 4, to 2, and then finally back to 1.

Part 1 of this unit was on creating graphs of parabolas and other polynomials, identifying their key features. Part 2 is more algebraic, focused on solving quadratic equations by hand and connecting the equations to the graphs more deeplyl. Today was a review of factoring and how that method of algebraic manipulation allows us to solve these equations.

Your project proposals are due at the start of class (access the full project details here). When you submit the proposal, please include all of your group members’ names in the filename (LastName.LastName.LastName.MidYearProjectProposal) before sharing it with me.

Today, we watched a Vsauce video about the classic trolley problem. The video deals with ethics, both of the problem itself and with creating a realistic experimental scenario to test how people would react in the situation posed. Tomorrow, we will have an organized and structured conversation about ethics ourselves. Please read over the Ethics Discussion Guidelines for details. At the start of class tomorrow, you will be assigned one of two stances to argue:

For some experiments, it is okay to lie to subjects in the name of proper experimental design and to preserve the integrity of what is being tested.

It is never okay to lie to subjects in an experiment, regardless of what is being tested.

Tomorrow, you’ll be asked to refer to the Vsauce video and the articles below to inform your arguments. Please read these articles, as well as take notes or even print them out so you can refer to them on Friday.

Left-click and drag a rectangle around any portion of the diagram to zoom in on that portion.

Right-click anywhere in the diagram to run a series of iterations at that value of a along the x-axis. The software will identify the magnitude of the cycle you’re looking at (though sometimes it makes a mistake, so look at the list as well!)

After some time to explore, we made a few observations about the general questions. The first is that cycles are born in two ways:

Bifurcations of “lower” cycles, or

Spontaneously arising from chaos

This suggests a certain ordering of cycles: the 2-cycle comes first, then the 4-cycle, the 8-cycle, the 16-cycle, and so on for powers of 2. Eventually, these cycles become so large that the system becomes chaotic. But this only explains cycles that are powers of 2. What about any other? For this, please read this article about the Feigenbaum plot, its constant, and the ordering of cycles.

You should have decided on an idea for your project today and started working on your proposal. That will be due at the start of class tomorrow, and I’ll give you about 10 minutes to discuss it when you get in tomorrow. Include all of your group members’ names in the filename (LastName.LastName.LastName.MidYearProjectProposal) before sharing it with me.

On Friday, we will be having an important discussion about experimental ethics. It will be a highly organized and structured conversation that will start with the ethics of lying to participants, but will proceed organically from there. Please read over the Ethics Discussion Guidelines for details.

At the start of class tomorrow, you will be assigned one of two stances to argue:

For some experiments, it is okay to lie to subjects in the name of proper experimental design and to preserve the integrity of what is being tested.

It is never okay to lie to subjects in an experiment, regardless of what is being tested.

We will watch a video about one controversial experiment conducted by the YouTube channel Vsauce, and on Friday you’ll be asked to use that video and the articles below to inform your arguments. You may want to start reading these now, as well as take notes or even print them out so you can refer to them on Friday.

We explored cycles a bit more in class today, noting that if {p,q} are the two values of a 2-cycle, then by definition f (p) = q and f (q) = p. Put another way, f (f (p)) = p and f (f (q)) = q. What’s important to note, though, is that the reverse is not true. If f (f (n)) = n, then n could be a member of a two cycle (n,m,n,m,n,m,…) or it could just be a fixed point: (n,n,n,n,n,n,…)

One important takeaway from this is that the first is that the local slope (Instantaneous Rate of Change) to all points of a 2-cycle, or indeed any cycle, is always the same. As a result, all points of a cycle will pass an IRC of -1 simultaneously, meaning all points of a cycle bifurcate simultaneously. This (and the Chain Rule from calculus) served as our proof of Question 2 posed early in our exploration of this pattern.

The implication of this is clear: after a = 3.0, the pattern of the orbit of the logistic function bifurcates to a two cycle. By a = 3.5 (more specifically at a = 3.449490…), it has bifurcated again to a four-cycle. Soon after (at a = 3.544090…), it bifurcates yet again to an eight-cycle, then to a 16-cycle (at a = 3.564407…), a 32-cycle (at a = 3.568759…), and so on. These bifurcations happen over increasingly shorter intervals for values of a. The fascinating thing is that the ratio between these intervals approaches a constant value, approximately 4.669202…, known as Feigenbaum’s Constant (this link goes to a Numberphile video about this idea that we watched a portion of in class today; the picture of orbit destinations that we have been drawing is known as the Feigenbaum Plot, which we will look at in detail tomorrow).

Today, you formed your groups for the Midterm Project and started brainstorming ideas for projects. By the end of the period, I want you to submit a Google Document to me with 2-3 “We wonder” statements that show me what you are considering for your research question.

Each statement statement must start with the phrase “We wonder…” Where you go after that is up to you, for example “We wonder if…” “We wonder why…” “We wonder whether…” For now, it should be fairly broad, you’ll make them more specific later. See the pre-break post for ideas.

What you should avoid is “We wonder what will happen when…” That is a method-forward, not question-forward.

We did some more research in the logistic map and the catalog of behavior we’ve been using for the past few days. We’ve answered the first question from yesterday’s list: the first split happens at precisely a = 3.0. With a proof in class, we showed that this is because for a = 3.0, the slope of the curve at the value of the fixed point (which happens to precisely equal 2/3) is exactly -1. Any value of a less than 3.0 will have a slope at its fixed point that is shallower than -1, and as a result the fixed point is an attractor. Any value of a greater than 3.0 will have a slope at its fixed point that is steeper than -1, and as a result the fixed point is a repeller. So when we notice that there’s a 2-cycle at a = 3.1, this is because the fixed point is a repeller, pushing the orbit to the 2-cycle. This 2-cycle exists for all values of 3.0 < a< 3.1.

We also have seen some progress on question 3, finding a 5-cycle at a = 3.906, a 7-cycle at a = 3.702, and our very first 3-cycle at a = 3.83. A follow-up question we could ask here: Can a cycle of any length be found for a > 3.6?

We still haven’t answered question 2: whether or not both points of the 2-cycle at a = 3.4 split simultaneously to form the four cycle found at a = 3.5, or if, for the briefest of moments, one point splits before the other and we can find another 3-cycle somewhere between 3.4 < a < 3.5. Your homework this break is to investigate this some more.